G-topological spaces and locales

Question

Consider the following generalization of topological spaces:

Definition: Let $X$ be a set. A G-topology on $X$ is given by certain distinguished subsets $U \subset X$, called admissible open subsets, as well as for each admissible open subset $U \subset X$ a system of certain distinguished set-theoretic coverings $U = \bigcup_{i \in I} U_i$ by other admissible open subsets $U_i \subset X$, called admissible coverings, in such a way that the following axioms are satisfied:

1. The intersection $U \cap V$ of two admissible open subsets $U, V \subset X$ is again admissible open.
2. For each admissible open subset $U \subset X$, the trivial covering $\{U\}$ of $U$ is admissible.
3. If $U$ is an admissible open subset of $X$, $\{U_i\}_{i \in I}$ is an admissible covering of $U$ and for each $i \in I$ the family $\{V_{ij}\}_{j \in J_i}$ is an admissible covering of $U_i$, then the family $\{V_{ij}\}_{i \in I, j \in J_i}$ is an admissible covering of $U$.
4. If $U, V \subset X$ are two admissible open subsets of $X$ and $V \subset U$, and if $\{U_i\}_{i \in I}$ is an admissible covering of $U$, then the family $\{V \cap U_i\}_{i \in I}$ is an admissible covering of $V$.

The admissible open subsets, ordered by inclusion, form a partially ordered set and hence a category. The covering data gives rise in an obvious way to a Grothendieck pre-topology and hence to a Grothendieck topos $\mathrm{Sh}(X)$ of sheaves on $X$. G-topological spaces appear prominently in rigid analytic geometry. The basic theory of G-topological spaces and sheaves on them is developed for instance in [BGR84, Sec. 9.1 and Sec. 9.2].

I know from [MLM92, Thm. 1 in Sec. IX.5] that a site whose underlying category is a poset gives rise to a localic topos, i.e. we must have $\mathrm{Sh}(X) = \mathrm{Sh}(X_{\mathrm{loc}})$ for some (uniquely determined) locale $X_{\mathrm{loc}}$. I also know that $X_{\mathrm{loc}}$ can be described as the locale of subobjects in the terminal object of $\mathrm{Sh}(X)$. However I wonder:

Questions: Can the locale $X_{\mathrm{loc}}$ be described more explicitly? Is it spacial? Is there a topology on the set $X$ such that $X_{\mathrm{loc}}$ is given by the open subsets of this topology? (I think that this is not in general the case.) Is there a topological space $X'$ satisfying $\mathrm{Sh}(X') = \mathrm{Sh}(X)$ and a G-topology on the underlying set of $X'$ together with a natural map $\beta \colon X \to X'$ of G-topological spaces, such that the operation $\beta^{-1}$ identifies the admissible open subsets of $X'$ with those of $X$ and the same holds for admissible coverings? I think that this is the case in the second example below.

Edit: Here is another remark: Any admissible open subset $U \hookrightarrow X$ gives rise to a morphism of topoi $\mathrm{Sh}(U) \to \mathrm{Sh}(X)$ and hence (by [MLM92, Prop. 2 in Sec. IX.5]) to a morphism $U_{\mathrm{loc}} \to X_{\mathrm{loc}}$. I think that by [MLM92, Prop. 5 (ii) in Sec. IX.5] this map is an embedding of locales, so both the ordered set of admissible open subsets of $X$ and the frame of opens of $X_{\mathrm{loc}}$ should embed in the frame of sublocales of $X_{\mathrm{loc}}$. Can we describe the relation between the two within this bigger poset?

For a concrete example, consider the set of real numbers $\mathbb{R}$ an consider the G-topological space with underlying set $\mathbb{R}$ where

• a subset $I \subset \mathbb{R}$ is admissible open if it is a compact interval, and
• a covering $I = \bigcup_{\alpha \in A} I_\alpha$ is admissible if $A$ is finite.

In this case the category of sheaves agrees (I think) with the category of sheaves on the ordinary topological space $\mathbb{R}$ with its Euclidean topology. If necessary, I can provide a proof.

For another example (which is actually quite similar), let $\mathrm{Sp}(A)$ be the rigid analytic space associated to a $K$-affinoid algebra $A$ where $K$ is a non-archimedean field. It is a G-topological space whose points are the maximal ideals of $A$, whose admissible open subsets are the affinoid domains and whose admissible coverings are the finite ones. Then the sheaves on $\mathrm{Sp}(A)$ agree with sheaves on $\mathscr{M}(A)$, the Berkovich space associated to $A$ with its Berkovich topology. There is a canonical map $\mathrm{Sp}(A) \to \mathscr{M}(A)$ identifying the affinoid domains of $\mathrm{Sp}(A)$ with the affinoid domains of $\mathscr{M}(A)$ in the Berkovich sense. As remarked by Piotr Achinger below, the Berkovich spectrum should be replaced with the Huber spectrum here.

[BGR84] S. Bosch, U. Güntzer and Reinhold Remmert, Non-archimedean analysis. Berlin etc.: Springer-Verlag (1984; Zbl 0539.14017)

[MLM92] S. Mac Lane and I. Moerdijk, Sheaves in geometry and logic: a first introduction to topos theory. New York etc.: Springer-Verlag (1992; Zbl 0822.18001)

Answer 1

Let us first try to describe the locale more explicitly. This will allow us to answer the other questions. As reference I will use https://www.math.ias.edu/~lurie/278xnotes/Lecture15-Spaces.pdf.

First, let us describe what the datum of an open in the locale $L$ associated to a G-topological space $X$ amounts to. As you noted, it is a subobject of the terminal object in the topos. This terminal object is determined by what admissible opens are sent to a point and which are sent to the empty set.

We observe that such subobjects can be identified with collections of admissible opens closed under taking subsets and coverings. Philosophically: The open is the collection of admissible opens lying within it.

Using definition 1 in the reference we see that points of the locale can be identified with opens as above satisfying the further conditions:

• The complement is closed under finite intersections
• It is prime, i.e. it is proper and if it is the meet of two opens it already agrees with one of them

Think of this as the set of admissible opens not containing a specific point x. We will identify an open or a point with the corresponding collection of admissible opens.

Every open in the locale $\mathcal{U}(Pt(L))$ comes from an an open $U$ of $L$. However, a lot of opens might be identified. Namely the ones containing the same points. In order to show $\mathcal{U}(Pt(L)) \cong L$ it suffices to see that intersection of points containing $U$ is again $U$.

Without the following assumption the conclusion is in general false: Every admissible covering is finite.

Let V be an admissible not contained in $U$. We need to find a point containing $U$, but not $V$. Let $Y$ be the set of collections of opens containing $U$, but not $V$. This is closed under directed unions. Here the assumption comes in, as it implies that the union as elements of the power set of admissible opens is already open. By Zorn's lemma there is a maximal element of $Y$. Let us call it $P$. We need to show it's a point.

First we see that $P$ is prime. It is proper since it doesn't contain $V$. Furthemore, when written as the intersection of two opens one of them has to be in $Y$. By maximality of $P$ it already has to be $P$.

we need to show that for $E_1, E_2 \notin P$ we have $E_1 \cap E_2 \notin P$. Since $P$ is maximal in $Y$ we may any of the $E_i$ together with the elements of $P$ to cover $V$. By intersecting the two coverings we see that $E_1 \cap E_2$ together with elements of $P$ covers $V$. Hence, $E_1 \cap E_2 \notin P$. This shows the claim under the assumption.

Without the assumption $L$ does not necessarily come from a topological space. For instance Example 18 from the reference can be seen as coming from a G-topological space where admissible coverings need not be finite. Namely $X = [0,1]$. The admissible opens are the measurable subsets. And finally admissible coverings are countable collections of maps that are jointly surjective up to null set.